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 {\Large \bf List of changes}

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 {\Large \bf paper 2553-F}

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 {\Large \sf Hilbert-Schmidt measure of pairwise quantum  discord in three qubit $X$ states}\\
   {\large \sf by M. Daoud et al}


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\vspace{1cm}

\noindent Dear editor,

\noindent We revised the paper according the  referee's comments. We added all his suggested improvements.

\noindent  We would like to thank the referee for his constructive remarks.\\



\textcolor{red}{\bf Concerning the comment:}


But when preparing
the final version of the manuscript, the authors may consider to include
some more explicit details on the applications and on the physical
implications of their choices of the X-states, to make it easier to read.


\textcolor{green}{\bf our reply:}

In the beginning of the second section, we added a short paragraph

\textcolor[rgb]{0.98,0.00,0.00}{ $X$ states of two qubits have already found applications in many studies of
entanglement and discord [12-27].
%As mentioned in the introduction, the interest in generalized $X$-states is motivated by the fact that they cover
%many different states of interest in quantum information such as ${\rm W}$ and
%${\rm GHZ}$ and Dicke states. The generalized $X$ states are of paramount importance
%in investigating  quantum correlations for a collection of spin-$1/2$ particles possessing particle exchange symmetry. For instance, the reduced %density matrices of multipartite Schr\"odinger cat states, which  are invariant under permutation symmetry, are  $X$ structured
% operators  (for more details, see the reference \cite{Seddik}). Completely symmetric systems, including Dicke states, are relevant in many %experimental situations such as spin squeezing . Indeed, spin squeezed state may have potential applications in atomic interferometers and high atomic %clocks (see \cite{wang+sanders}) . Also, The multi-qubit $X$ states arise naturally in describing the dynamics tripartite quantum spin states %interacting with a large environment \cite{J.Zhou}.
$\cdots$ $\cdots$  $\cdots$  This is of crucial importance in analyzing the decoherence effects induced by the environment in such systems.}

 quoting some references to give an overview of the relevance of the $X$ states from a theoretical as well experimental point of views.\\




\textcolor{red}{\bf Concerning the comment:}

More careful introductions of the notions of quantum discord and its
measures and on the monogamy property may also make the paper easier to
follow.


\textcolor{green}{\bf our reply:}


In the beginning of the third section, we discuss briefly the concept of quantum discord quoting the related pioneer works. Thus we added
the following paragraph:

\textcolor[rgb]{0.98,0.00,0.00}{A bipartite quantum system exhibits quantum correlation if its two subsystems contain more information than taken separately
%. This concept is captured by
%the mutual information  $I(A:B)=H(A)+H(B) - H(A,B)$
%where $A$ and $B$ are random variables. In classical information theory $H(.)$ stands
% for the Shannon entropy $H(p)= - \sum_i p_i\log p_i$ where $p=(p_1, p_2, \cdots)$ is the probability distribution. For a quantum density matrix %$\rho$,  it is denoted by the von Neumann entropy  $H(\rho)= -{\rm Tr}\rho\log \rho$. In the classical case, an equivalent expression for the mutual %information is given by   $I(A:B)=H(A)- H(A|B)$ where $H(A|B)$ is the Shannon entropy of
%$A$ conditioned on the measurement outcome of $B$. In the quantum case, the two expression are different and the difference defines the so-called %quantum discord \cite{Ollivier-PRL88-2001,Vedral-et-al}. The von Neumann entropy-type quantum discord involves complicated optimization procedures. In %the literature there are few examples for which closed analytical expressions for quantum discord were obtained (see \cite{Vedral-RMP-2012}). %Alternatively, distance-type quantifiers of quantum discord have been considered. This is essentially motivated by their presumably simple evaluation %in comparison with the original quantum discord definition. Several distances are possible (trace distance, Bures distance, ...)
%with their own advantages and drawbacks.
$\cdots$ $\cdots$ $\cdots$
In this paper we shall especially consider the
geometric discord variant based on Hilbert-Schmidt norm [10].}


Similarly, concerning the notion of the monogamy, we added (in the beginning of section 4):

\textcolor[rgb]{0.98,0.00,0.00}{The quantum correlation can be transferred between the components of a quantum system comprising many parties. This shareability
%is however subject to the monogamy relation which writes for a three qubits system as
%$$ Q_{1|23} \geq Q_{1|2} + Q_{2|3}$$
%where $Q$ denotes a measure of pairwise quantum correlation in the system. This inequality means that the amount of quantum correlation shared between %the qubits 1 and 2 restricts the possible amount of quantum correlation between the qubits 2 and 3 so that the sum is always less than the total
$\cdots$ $\cdots$ $\cdots$
bipartite correlation between the qubit 1 and the subsystem containing the qubits 2 and 3.}




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\begin{document}

\title{Algebraic characterization of $X$-states in quantum information}

\author{A.\ R.\  P. Rau$^{*}$}
\affiliation{Department of Physics and Astronomy, Louisiana State University,
Baton Rouge, Louisiana 70803-4001}

%\date{\today}

\begin{abstract}

A class of two-qubit states called $X$-states are increasingly being used to discuss entanglement and other quantum correlations in the field of quantum information. Maximally entangled Bell states and ``Werner" states are subsets of them. Apart from being so named because their density matrix looks like the letter X, there is not as yet any characterization of them. The $su(2) \times su(2) \times u(1)$ subalgebra of the full $su(4)$ algebra of two qubits is pointed out as the underlying invariance of this class of states. $X$-states are a seven-parameter family associated with this subalgebra of seven operators. This recognition provides a route to preparing such states and also a convenient algebraic procedure for analytically calculating their properties. At the same time, it points to other groups of seven-parameter states that, while not at first sight appearing similar, are also invariant under the same subalgebra. And it opens the way to analyzing invariant states of other subalgebras in bipartite systems.

\end{abstract}

\pacs{03.67.-a, 02.20.Sv, 03.67.Bg, 03.67.Mn, 03.65.Ud}

\maketitle

\section{Introduction}

Increasingly in the field of quantum information, aspects of entanglement \cite{ref1}, and of other quantum correlations such as, for instance, ``quantum discord" \cite{ref2}, between two qubits have been described for a class of pure and mixed states that have come to be called ``$X$-states" \cite{ref3}. Although their use goes back further \cite{ref4}, they were so named in \cite{ref3} because of the visual appearance of the density matrix, that it looks like the letter in the alphabet:

\begin{equation}
\rho =  \left(
\begin{array}{cccc}
\rho_{11} & 0 & 0 & \rho_{14} \\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{32} & \rho_{33} & 0 \\
\rho_{41} & 0 & 0 & \rho_{44}
\end{array}
\right).
\label{eqn1}
\end{equation}

Non-zero entries occur only along the diagonal and anti-diagonal. Many calculations of entanglement and other properties \cite{ref4,ref5}, and their evolution under unitary or dissipative processes \cite{ref6}, can be carried out analytically for such states which make them appealing objects for study. Many other states of interest, such as the maximally entangled Bell states \cite{ref1} and ``Werner" states \cite{ref7}, are a sub-class of $X$-states, lending further importance to their study.

Yet, no firmer definition has been given of what makes a pure or mixed system an $X$-state. This Letter provides such a definition in terms of their invariance properties, that a particular symmetry group or algebra underlies them. Such an identification of an underlying symmetry helps to explain the analytical results while at the same time providing a well defined procedure for their preparation. Recognizing the symmetry also makes computations involving such states, such as unitary operations on them or evaluating concurrence or other measures of entanglement, straightforward and easily tractable. And, finally, the symmetry also opens the way for constructing other density matrices which may not visually appear as X but are nevertheless similar, states of a different rendering of the same algebraic symmetry. Since they differ in entanglement and separability considerations, they may prove useful for study.

\section{The subalgebra of $X$-states}

Positivity and other standard requirements of any density matrix make the $X$-states shown in Eq.~(\ref{eqn1}) a seven-parameter family. The diagonal elements of the density matrix are real so that, along with the trace being fixed at 1, three real parameters describe those diagonal entries. Hermiticity to guarantee real eigenvalues reduces the off-diagonal entries to two complex (say $\rho_{14}$ and $\rho_{23}$, with $\rho_{41}$ and $\rho_{32}$ their respective complex conjugates) or four real parameters for the total of seven real parameters.

The full two qubit system has the symmetry of the SU(4) group and its algebra $su(4)$. Fifteen operators, most conveniently rendered as fifteen linearly independent 4 $\times$ 4 matrices or as Pauli spinors/matrices of the two spins, together with the unit matrix, provide a complete description of the general system. There are, however, several subalgebras of $su(4)$. A series of recent papers have provided a geometrical description of their states and operators \cite{ref8,ref9,ref10,ref11}. In particular, one subalgebra, $su(2) \times su(2) \times u(1)$, of seven operators or matrices occurs in many physical systems in quantum optics and quantum information \cite{ref8,ref9}. This Letter presents them as the invariance set of the $X$-states.

Inspection of the explicit 4 $\times$ 4 matrices in a standard basis for two spins, $(|\uparrow \uparrow
\rangle, |\uparrow \downarrow \rangle, |\downarrow \uparrow \rangle,
|\downarrow \downarrow \rangle)$, is instructive \cite{ref8,ref9,ref12,ref13} and points immediately to sets of seven of them with the same structure of eight zeroes in the same positions as in Eq.~(\ref{eqn1}). That is, these are operators that do not mix the 1-4 and 2-3 subspaces of the density matrix. Combined with the observation that such a set of seven matrices closes under multiplication, it is immediate that they will carry $X$-states into each other, that they preserve the X structure. For this purpose, both the Lie algebra aspect that the seven operators close under commutation and their Clifford algebraic structure that they close under multiplication are important. Indeed, explicit rendering of the fifteen operators in terms of two Pauli spinors called $\vec{\sigma}$ and $\vec{\tau}$, together with the familiar multiplication rule $\sigma_i \sigma_j = \delta_{ij} + i \epsilon_{ijk} \sigma_k, i, j, k = 1-3$, where $\epsilon_{ijk}$ is the completely antisymmetric symbol and repeated indices are summed, is very useful for operations with them.

There are many such sets of seven operators/matrices constituting the $su(2) \times su(2) \times u(1)$ subalgebra \cite{ref8,ref9,ref11}. In each of them, one operator, the $u(1)$ element, commutes with all six of the others which themselves can be further subdivided as shown in \cite{ref8} into two sets of ``pseudospins", two sets of three which obey commutation relations of angular momentum within each set while all three of one set commute with all three of the other. Any one of the fifteen operators can serve as the commuting element because, as shown in a table in \cite{ref9}, each row has six zeroes so that each identifies such a $su(2) \times su(2) \times u(1)$ set. There are, therefore, fifteen non-equivalent such subalgebras.

We will designate such a set by $\{X_i\}, i=1, 2, \ldots, 7$, with $X_1$ the commuting element. One such is $(X_1 =\sigma_z \tau_z, X_2=\sigma_y \tau_x, X_3=\tau_z, X_4=-\sigma_y\tau_y, X_5=\sigma_x\tau_y, X_6=\sigma_z, X_7=\sigma_x\tau_x)$. This is the same set that occurs in the CNOT quantum logic gate constructed out of two Josephson junctions and was extensively studied in that context \cite{ref8}. It was also pointed out that it occurs in nuclear magnetic resonance when each spin is in an external magnetic field in the $z$-direction while being coupled to each other through scalar coupling $\vec{\sigma} \cdot \vec{\tau}$ and ``cross-coherences" $\sigma_x\tau_y$ and $\sigma_y\tau_x$. But a different choice for the commuting element $X_1$ gives another such subalgebra, and we will return to this in section IV. Each $X_i$ is traceless, Hermitian, and unitary, and its square is unity so that the eigenvalues are ($\pm 1, \pm 1$).

With any such set, $\{X_i\}$, the density matrix that remains invariant under their operations can be rendered as a linear superposition of them,

\begin{equation}
\rho =(I + \Sigma_i g_i X_i)/4,
\label{eqn2}
\end{equation}
in analogy to that for a single spin, $(I+\Sigma_i g_i \sigma_i)/2$. The seven real coefficients $g_i$ in Eq.~(\ref{eqn2}) parametrize $X$-states and are equivalent to the seven parameters in the density matrix in Eq.~(\ref{eqn1}):

\begin{eqnarray}
g_1 & = & (\rho_{11} +\rho_{44}) -(\rho_{22} +\rho_{33}), \nonumber \\
g_2 & = & 2i(\rho_{14}-\rho_{41}+\rho_{32}-\rho_{23}), \nonumber \\
g_3 & = & (\rho_{11} -\rho_{44}) -(\rho_{22} -\rho_{33}), \nonumber \\
g_4 & = & 2(\rho_{14}+\rho_{41}-\rho_{32}-\rho_{23}), \nonumber \\
g_5 & = & 2i(\rho_{14}-\rho_{41}-\rho_{32}+\rho_{23}), \nonumber \\
g_6 & = & (\rho_{11} -\rho_{44}) +(\rho_{22} -\rho_{33}), \nonumber \\
g_7 & = & 2(\rho_{14}+\rho_{41}+\rho_{32}+\rho_{23}).
\label{eqn3}
\end{eqnarray}

The algebra of the seven $\{X_i\}$ is most conveniently captured by Fig. 1 as has recently been pointed out \cite{ref11}. This figure occurs in projective geometry as the ``Fano Plane" \cite{ref14} and also is used to represent the multiplication table for octonions \cite{ref15}. Arranging the seven operators at the vertices, mid-points of sides and in-center of an equilateral triangle, the seven lines shown (including the inscribed circle) each carry three points, providing the multiplication rule for those $\{X_i\}$. The notation of arrows is also borrowed from octonions except that unlike them which have all seven lines arrowed, the three internal verticals are not in Fig. 1. On those lines, all three operators mutually commute, so that the product of two gives the third regardless of order. On the four arrowed lines, the operators mutually anticommute so that the product of two gives $(\pm i)$ times the third, with plus (minus) signs along (against) the sense of the arrow. For this purpose, each line is regarded as a closed loop with a continuously circulating arrow. The central element commutes with all six of the others. For each of those, there is one ``conjugate" element with which it commutes and four with which it anticommutes. All of this can be read off by merely glancing at Fig. 1 and will provide simple rules for their manipulation in the next section.

\begin{figure}
\vspace{-.5in}
\scalebox{1.6}{\includegraphics[width=2.in]{Rau-X-Fig.eps}}
\vspace{-.4in}
\caption{The multiplication diagram for the seven operators that underlie $X$-states. Resembling the Fano Plane and the multiplication diagram for octonions, each operator stands on three lines, and each line, including the inscribed circle, has on it three operators. On the interior verticals, the product of any two operators gives the third, these objects commuting. On the remaining four lines, the operators anticommute, and the product of any two gives the third with a multiplicative $\pm i$, the plus (minus) depending on the direction of (along/against) the arrow \cite{ref11}.}
\end{figure}

\section{Manipulating density matrices with the elements of the subalgebra}

The Clifford algebra structure and its diagrammatic rendering in Fig. 1 makes operations on the density matrix of $X$-states very simple. Thus, $X_1 \rho X_1^{\dagger}$ leaves $\rho$ in Eq.~(\ref{eqn2}) unchanged. For any of the other six cases, $X_i \rho X_i^{\dagger}$, three coefficients in Eq.~(\ref{eqn2}) remain unchanged (those belonging to that $i$, its conjugate, and 1) while the other four are switched to their negative. As an example of another common operation which occurs, for instance, in the evaluation of ``concurrence" \cite{ref16} or ``quantum discord" \cite{ref2}, $\tilde{\rho} = \sigma_y \tau_y \rho^* \sigma_y \tau_y$ can also be written down from Eq.~(\ref{eqn2}) without any calculation. This involves $X_4$. In Fig. 1, this element is connected to $i=2,3,5,6$ by arrows so that those coefficients $g_i$ have their signs changed by this operation. But the additional complex conjugation involved changes signs again for $i=2,5$ whose $X_i$ in Eq.~(\ref{eqn2}) is pure imaginary. As a result, $\tilde{\rho}$ differs from $\rho$ in Eq.~(\ref{eqn2}) by just flipping the sign of $i=3,6$. In like manner, all such operations become almost automated. And, in evaluating the concurrence through the eigenvalues of $\rho \tilde{\rho}$, with the products of $\{X_i\}$ staying within the algebra and again easily written down with reference to Fig. 1, this construct is again a linear combination of the form of Eq.~(\ref{eqn2}). The square roots of the four eigenvalues turn out to be

\begin{eqnarray}
\!\!& & \!\!\frac{1}{2}[\sqrt{(1\!+ \! g_1)^2 \! - \! (g_3 \! + \! g_6)^2} \pm\!\! \sqrt{((g_2 \! + \! g_5)^2 \! +\! (g_4 \! + \!g_7)^2}] \nonumber \\
\!\!& & \!\! \frac{1}{2}[\sqrt{(1\! - \! g_1)^2 \! - \! (g_3 \! - \! g_6)^2} \pm\!\! \sqrt{((g_2 \! - \! g_5)^2 \!+\! (g_4 \! -\! g_7)^2}].
\label{eqn4}
\end{eqnarray}
Note the appearance of the combinations of conjugate pairs 2-5, 3-6 and 4-7 of Fig. 1 (and, of course, 1 with the unit element). Arranging the quantities in Eq.~(\ref{eqn4}) in decreasing order and subtracting the sum of the last three from the highest provides the concurrence for any $X$-state.

A prescription for creating $X$-states from a general density matrix can also be easily provided. Each of the fifteen operators of $su(4)$ commutes with six and anticommutes with eight of the others \cite{ref8,ref9}. Thus, $X_1$ which commutes with the six in the $su(2) \times su(2) \times u(1)$ set necessarily anticommutes with all the eight left out of this set. As a result, in a general density matrix $\rho$ of two qubits with all elements nonzero, action by this element, $X_1 \rho X_1^{\dagger}$, changes the signs of the entries in those eight positions which have zeroes in Eq.~(\ref{eqn1}), while leaving other coefficients unchanged. A a result, starting from any general $\rho$, adding to it $X_1 \rho X_1^{\dagger}$ will generate an $X$-state. The action of $X_1(=\sigma_z \tau_z)$ is, of course, the simultaneous unitary rotation through $\pi$ of both spins.

In the dynamics of two qubits as well, such as in entanglement evolution, including in dissipative processes, the $X$-structure of the states is preserved when the Kraus or other operators involved are expressible in terms of the set $\{X_i\}$.

\section{Other classes of ``$X$-states"}

The identification above of the ``standard" (as commonly used) $X$-states  with the element $X_1=\sigma_z\tau_z$ immediately points to several other groups of states having the same characteristic of an invariant algebra of $su(2) \times su(2) \times u(1)$ but with different choices for $X_1$. One, for instance, is to choose for this element $\sigma_z$. The other six members are $(\vec{\tau}, \sigma_z \vec{\tau})$. All these matrices and the resulting density matrix that is invariant in form under their operations are now block diagonal $4 \times 4$ matrices, with all eight elements  in the two off-diagonal blocks  zero. While not looking like the X in Eq.~(\ref{eqn1}), they now stand for a decoupling of the 1-2 and 3-4 subspaces, different from that in Eq.~(\ref{eqn1}). In terms of the basis states, they can now be very different in separability and entanglement properties, grouping $(|\uparrow \uparrow
\rangle, |\uparrow \downarrow \rangle)$ and $ (|\downarrow \uparrow \rangle,
|\downarrow \downarrow \rangle) $ together instead of $(|\uparrow \uparrow
\rangle, |\downarrow \downarrow \rangle) $ and $(|\uparrow \downarrow \rangle, |\downarrow \uparrow \rangle)$. Nevertheless, algebraically they are also invariant sets of an $su(2) \times su(2) \times u(1)$ algebra.

Even more interestingly, an $X$-state need not have any zeroes in its density matrix! Thus, the choice, $(X_1 =\sigma_x \tau_x, X_2=\sigma_z \tau_y, X_3=\tau_x, X_4=-\sigma_z\tau_z, X_5=\sigma_y\tau_z, X_6=\sigma_x, X_7=\sigma_y\tau_y)$, built on commuting element $X_1=\sigma_x\tau_x$ is equally valid as an $X$-state with the same invariance algebra, although its density matrix has no zero entries:

\begin{equation}
\rho = \! \frac{1}{4}I \! + \! \frac{1}{4}\left(
\begin{array}{cccc}
-g_4+g_6 & g_3-ig_2 & -ig_5 & g_1-g_7 \\
g_3+ig_2 & g_4+g_6 & g_1+g_7 & ig_5 \\
ig_5 & g_1+g_7 & g_4-g_6 & g_3+ig_2 \\
g_1-g_7 & -ig_5 & g_3-ig_2 & -g_4-g_6
\end{array}
\right).
\label{eqn5}
\end{equation}
Its preparation, by adding to $\rho$ a transformation under $X_1 (=\sigma_x \tau_x)$, and its manipulation or calculation of concurrence and other properties, all proceed as in Section III. Indeed, as can be seen from the choice of the two sets of $\{X_i\}$, it differs from the ``standard" set by a cyclic permutation of the indices $(x, y, z)$ which means two $\pi/2$ rotations of axes, first clockwise with respect to $y$ and then counter-clockwise with respect to an intermediate $z$ axis. Alternatively, this can be represented as the application of $\pi$ pulses to the spins. Clearly, the physics is unchanged with mere change of bases.

Entanglement and other investigations of these other sets will be of interest. A natural extension for further investigation is to analyze similarly sets of invariant states of other subalgebras of $su(4)$ such as $su(3)$ and $so(5)$ \cite{ref10}. The recognition of invariant sets opens a new window into such studies, focusing on what is essential and what are simply changes in bases and representations.


I thank Mr. Mazhar Ali, and Drs. Gernot Alber and Joseph Renes for discussions and for their hospitality at the Technical University, Darmstadt, during this work. This work was supported by the Alexander von Humboldt Foundation.

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\bibitem[*]{} Email: arau@phys.lsu.edu

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\bibitem{ref3} T. Yu and J. H. Eberly, Quantum Inform. and Comput. {\bf 7}, 459 (2007); Phys. Rev. Lett. {\bf 93}, 140404 (2004); {\it ibid} {\bf 97}, 140403 (2006).

\bibitem{ref4} See, for instance, S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, Phys. Rev. Lett. {\bf 87}, 050401 (2001); G. L, Kamta and A. F. Starace, {\it ibid} {\bf 88}, 107901 (2002); J. S. Pratt, {\it ibid} {\bf 93}, 237205 (2004); S. J. Gu, G. S. Tian, and H. Q. Lin, Phys. Rev. A {\bf 71}, 052322 (2005); J. Wang, H. Batelaan, J. Podany, and A. F. Starace, J. Phys. B {\bf 39}, 4343 (2006).

\bibitem{ref5} R. Dillenschneider, Phys. Rev. B {\bf 78}, 22413 (2008); S. Luo, Phys. Rev. A {\bf 77}, 042303 (2008); M. S. Sarandy, arXiv: 0905.1347; T. Werlang, S. Souza, F. F. Fanchini, and C. J. Villas Boas, arXiv: 0905.3376.

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\bibitem{ref10} D. Uskov and A. R. P. Rau, Phys. Rev. A {\bf 74}, 030304 (R) (2005); {\it ibid} {\bf 78}, 022331 (2008); Sai Vinjanampathy and A. R. P. Rau, arXiv: 0906.1259.

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\bibitem{ref12} F. J. M. van de Ven and C. W. Hilbers, J. Magn. Reson. {\bf 54}, 512 (1983).

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\bibitem{ref14} T. Beth, D. Jungnickel, and H. Lenz, {\it Design Theory}, Vols. 1 and 2, Encyclopaedia of Mathematics, Vol. 69 (Bibl. Inst., Z\"{u}rich, 1985; Cambridge University Press, 1993).

\bibitem{ref15} See, for instance, G. M. Dixon, {\it Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics}, Vol. 290 of Mathematics and its Applications (Kluwer, Dordrecht, 1994); L. E. Dickson, Ann. Of Math. {\bf 20}, 155 (1919); H. S. M. Coxeter, Duke Math. J. {\bf  13}, 561 (1946); J. C. Baez, Bull. Am. Math. Soc. {\bf 39}, 145 (2002) and www.math.ucr.edu/home/baez/octonions/

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\end{thebibliography}

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The manuscript under review deals with  the quantum resources from the aspect of quantum state superposition.  From this point of view the authors claim that the quantum ressources can be categorized as superpositions existing in different parts in a multipartite quantum system.  In my opinion, the results presented here are of some interest and probably worth publishing.  However, some aspects of the manuscript are not totally convincing. The authors present their results without giving any physical or informational meaning of their correlation quantifier. Also, It is not clear the relation with entanglement of formation and quantum discord. The authors should clarify this point before resubmitting their paper.


However, the generation of these Bell cat-states seems to be
3
very difficult. The authors explain that this can be done using a coherent cat-state (superposition of
two coherent states) and they mention some theoretical proposals to generate such cat states, Refs.
[45,46,47,48] of the manuscript. My point is: as the whole work depends on the generation of coherent
cat-states, I would like to see a more detailed explanation about how it can be generated and what it
is the best efficiency one can have with the present day technology. Also, how this efficiency is affected
for highly excited states (i.e, states with a high mean number of photons)?
   

REFEREE REPORTS:
Referee: 1

Recommendation: Reject (average work that adds little new knowledge to the field)

Comments:
In my opinion, the main results of this work are simple reformulations of the separability conditions in the case of block-Toeplitz matrices. As such, they appear evident to me and not worth publishing.

More specifically:

1. Theorem 1  (page 4 of the manuscript) assumes that all blocks of the block-Toeplitz matrix commute. Then it is obvious to diagonalize all of them simultaneously and re-write the original matrix in the corresponding basis. Then it will separate into a sum of Toeplitz matrices. The theorem states that if those Toeplitz matrices are all positive semidefinite, then this decomposition is the same as in the definition of a separable state. This appears to me a trivial application of the separability definition.

2. Remark 3 (page 7 of the manuscript) gives a condition for the absence of quantum discord, which is a trivial application of the Theorem 2 (borrowed from ref [14]). The condition (40) mentioned in Remark 3 is simply a way to write the condition that matrices involved in the decomposition commute, as required by Theorem 2.

In my understanding, theorem 1 and remark 3 are supposed to be the main results of the paper. Both of them appear trivial to me (modulo Theorem 2, which is not trivial, but whose proof is not part of this paper). So, while the paper seems to be correct, I think it adds little knowledge to the field and should not be published.

CONFIDENTIAL COMMENTS TO THE EDITOR(S):
I have certain expertise in Toeplitz matrices and quantum entenglement, but I have little expertise in the theory of separable states. My negative report is based on the analysis of the proofs presented in the paper, which I find simple reformulations of definitions, together with previously existing results. It may be worth asking for an independent opinion from an expert specifically in the theory of separable states.

Additional Questions:
Originality: 5
Scientific Rigour: 10
Significance: 2

What impact will this article have in the field?: Low

Has the author made reference to the most recent and most appropriate work? <br>(If no, please provide references in the Comments to the author(s) section.): Yes

Is the article well written and the work clearly communicated? <br>(Please note we will edit upon acceptance and correct any grammatical or typographical errors.): Yes

Is this article better suited to a more specialised journal?<br>(If yes, please indicate in the Confidential comments to the editor(s) section which journal it would be suitable for.): No

Do you think your quality assessment scores will change if the author satisfactorily implements your recommendations?<br>(If no, please explain why in the Confidential comments to the editor(s) section.): No

Is this article fully or partially in your area of expertise? <br>(If partially, please provide an explanation in the Confidential comments to the editor(s) section.): Partially in my area of expertise


Referee: 2

Recommendation: Accept

Comments:
This is an interesting work that links entanglement and quantum discord to Toeplitz matrices. By this the authors succeeded to construct explicitly separable decompositions which is a nontrivial task. The work is also well motivated and show interesting interdisciplinary results. Therefore I suggest the publication. However, there are extremely many typos (please, consider to use immediately spelling control within your LateX interpreter!!) and the English should be corrected (often articles are missing). See some (only few) examples below. The results presented in the article are enough to be considered as a publication, however, the authors may think about all the other types of entanglement such as multipartite entanglement, whether also such a connection can be found.

1.)Abstract: genreral ->general
2.) many times: symplex  ->simplex
3.) Page 5: propieties
4.) Line 50: In this section ...quantum discord BEHAVES WITHIN the class of Toeplitz matrices  STUDIED ...
5.) Line 55: let us remind here --> let us recall the  theorem ...
6.)Line 58: we are given with --> we have given
7.) Page 6:Line9: writtien
8.)  Line 16: thogeter
9.) Line 19: liketo
...
10.) References [7]: comma missing, sometimes the Journals are written out sometimes not,...
...

CONFIDENTIAL COMMENTS TO THE EDITORI(S):
I am an expert in entanglement theory and information science which this works links to but not on the Toeplitz matrices. I found the connection presented exciting, however, there English needs correction and there were many typos.

Additional Questions:
Originality: 7
Scientific Rigour: 8
Significance: 7

What impact will this article have in the field?: Medium

Has the author made reference to the most recent and most appropriate work? <br>(If no, please provide references in the Comments to the author(s) section.): Yes

Is the article well written and the work clearly communicated? <br>(Please note we will edit upon acceptance and correct any grammatical or typographical errors.): Yes

Is this article better suited to a more specialised journal?<br>(If yes, please indicate in the Confidential comments to the editor(s) section which journal it would be suitable for.): No

Do you think your quality assessment scores will change if the author satisfactorily implements your recommendations?<br>(If no, please explain why in the Confidential comments to the editor(s) section.): Yes

Is this article fully or partially in your area of expertise? <br>(If partially, please provide an explanation in the Confidential comments to the editor(s) section.): Partially in my area of expertise

Letter reference: IS10

